Properties

Label 22218bh
Number of curves $4$
Conductor $22218$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("bh1")
 
E.isogeny_class()
 

Elliptic curves in class 22218bh

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
22218.bh4 22218bh1 \([1, 0, 0, -316882, 111971828]\) \(-23771111713777/22848457968\) \(-3382391787572013552\) \([2]\) \(506880\) \(2.2516\) \(\Gamma_0(N)\)-optimal
22218.bh3 22218bh2 \([1, 0, 0, -5913702, 5533051680]\) \(154502321244119857/55101928644\) \(8157062992429104516\) \([2, 2]\) \(1013760\) \(2.5982\)  
22218.bh2 22218bh3 \([1, 0, 0, -6765392, 3834952158]\) \(231331938231569617/90942310746882\) \(13462725819128930848098\) \([2]\) \(2027520\) \(2.9447\)  
22218.bh1 22218bh4 \([1, 0, 0, -94611132, 354202649010]\) \(632678989847546725777/80515134\) \(11919129439644126\) \([2]\) \(2027520\) \(2.9447\)  

Rank

sage: E.rank()
 

The elliptic curves in class 22218bh have rank \(1\).

Complex multiplication

The elliptic curves in class 22218bh do not have complex multiplication.

Modular form 22218.2.a.bh

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{3} + q^{4} + 2 q^{5} + q^{6} + q^{7} + q^{8} + q^{9} + 2 q^{10} - 4 q^{11} + q^{12} + 2 q^{13} + q^{14} + 2 q^{15} + q^{16} - 6 q^{17} + q^{18} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

sage: E.isogeny_class().matrix()
 

The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with Cremona labels.